Checking Cases

Problems that look complicated can often be broken down into simpler scenarios that contain more information. Since the individual case is easier to consider, the whole problem becomes easier to solve.

Some questions may provide you with the different cases. Then your job will be to determine if each scenario is a solution or not. Other questions will require you to think of all possible scenarios. Then, keep in mind all the givens and all the restrictions, but most importantly, when listing all the possibilities, be systematic!

How many triples of positive integers are there such that $a \times b + c = 8$?

(A) $\ \ 4$
(B) $\ \ 8$
(C) $\ \ 12$
(D) $\ \ 16$
(E) $\ \ 20$

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The correct answer is: D

Solution:

If $a = 1$, then we can have $b = 1, 2 \ldots 7,$ which gives us 7 solutions.
If $a = 2$, then we can have $b = 1, 2,$ and $3,$ which gives us 3 solutions.
If $a = 3$, then we can have $b =1$ and $2,$ which gives us 2 solutions.
If $a =4, 5, 6$ or $7,$ we can only have $b = 1$. This gives us 4 solutions.

Hence, in total, there are $7 + 3 + 2 + 4 = 16$ solutions.

Thus, the answer is (D).

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Incorrect Choices:

(A), (B), (C), and (E)
These answers are just offered to confuse you.

If you got this problem wrong, you should review Algebraic Manipulations.

Points $X$ and $Y$ are two vertices of a regular hexagon. Line $l$ is the perpendicular bisector of $XY$. Which of the following statements could be true?

\(\begin{array}{r r l} & \text{I.} & \text{ Line}\ l\ \text{passes through another vertex of the hexagon.}\\
& \text{II.} & \text{ Line}\ l\ \text{is perpendicular to an edge of the hexagon.}\\
& \text{III.} & \text{ Line}\ l\ \text{passes through the center of the hexagon.}\\
\end{array}\)

(A)$\ \$ I only
(B)$\ \$ III only
(C)$\ \$ I and II only
(D)$\ \$ I and III only
(E)$\ \$ I, II, and III

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Correct Answer: E

Solution:

Let the vertices of the hexagon be labelled $A, B, C, D, E, F$.

I. The perpendicular bisector of $AC$ passes through $B$, so option I could be true.

II. The perpendicular bisector of $AB$ is perpendicular to edge $AB$, so option II could be true.

III. The perpendicular bisector of any 2 vertices will pass through the center of the hexagon. Option III is always true.

Hence, options I, II, and III could be true.

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Incorrect Choices:

(A), (B), (C), and (D)
See the solution for why these choices are wrong.

If you got this problem wrong, you should review Regular Polygons.